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Intercept theorem (Thales)

Missing length of a Thales configuration, or the parallelism test by the converse, with the exact-ratio sketch.

Parameters

A is the common vertex; B and D on one line, C and E on the other, (BC) // (DE).

Result

Length AE

7.5

Calculation
AE = AC × AD / AB = 3 × 5 / 2 = 7.5
Reduction ratio
0.4
ADBEC
Notes (2)

By the intercept theorem, AE = 7.5.

Converse: the points must be aligned in the same order (A, B, D on one side, A, C, E on the other).

Calculation, assumptions and sources

OverviewIntercept theorem: the fourth length when three are known (AB/AD = AC/AE, by cross-multiplication), and the converse, which tests whether two lines are parallel by comparing ratios.

Intercept theorem: the fourth length when three are known (AB/AD = AC/AE, by cross-multiplication), and the converse, which tests whether two lines are parallel by comparing ratios. The sketch keeps the ratios of the entered lengths, with the parallel highlighted.

Method & formulasAB/AD = AC/AE = BC/DE
AB/AD = AC/AE = BC/DEAE = AC × AD / ABconverse: equal ratios + same order ⇒ (BC) // (DE)

Two lines meeting at A, cut by the parallels (BC) and (DE): the three ratios are equal. Three lengths of one equality give the fourth by cross-multiplication; BC/DE links the parallels themselves.

Thales configuration
two lines meeting at A and two parallels cutting them: B, D on one, C, E on the other.
Reduction ratio
AB/AD, the factor taking triangle ADE to triangle ABC; above 1, it is an enlargement.
Converse
equal ratios prove parallelism, provided the points are aligned in the same order.
Butterfly configuration
B and C on the other side of A, triangles opposed at the vertex; the ratios stay equal.
Validity domainThe theorem holds in the Euclidean plane; uncertainties come from the measurements.

The theorem holds in the Euclidean plane; uncertainties come from the measurements. The converse compares the ratios with a relative tolerance, and a gap under one per mille is measurement uncertainty. Alignment and point order remain to be checked on the figure.

The alignment conditionBesides equal ratios, the converse requires A, B, D aligned on one side and A, C, E on the other, with B between A and D if and only if C is between A and E.

Besides equal ratios, the converse requires A, B, D aligned on one side and A, C, E on the other, with B between A and D if and only if C is between A and E.

Reading the resultA reduction ratio of 0.4 means triangle ABC is ADE reduced to 40%; every length, BC included, follows that factor.

A reduction ratio of 0.4 means triangle ABC is ADE reduced to 40%; every length, BC included, follows that factor.